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This is toy model: they are only considering motion in the $x$ direction. In 3-d a triatomic molecule has 9 modes of which 6 are zero frequency (3 rotations, and 3 translations) and the other 4 are …

The modes that grow with time correspond to the black hole absorbing radiation incoming from infinity. These modes have to exist because of time reversal symmetry: if there is a solution with radiat …

Yes horizons can vibrate. The resulting damped oscillations are called "quasi-normal modes." There is a large literature on them: see arXiv:gr-qc/9909058 for a review.

Elastic waves in an isotropic solid are of two types. Longitudinal in which the particles move backwards and forwards in the direction of propagation and transverse where they move at right angles to …

The "$n$" in the $J_n(\kappa r)$ refers to the number of nodes in the angular direction. A complete set of eigenfunctions of $-\nabla^2$ in ${\mathbb R}^2$ are $$ \psi_{n,\kappa}(r,\theta)= e^{in\th …

The $q_{nm}$ are the amplitudes of the normal modes of vibration of the plate with sides $2\pi a$, $2\pi b$. He is using the usual plate wave equation $$ \frac{\partial^2 y}{\partial t^2}= D \left(\f …

For a fixed orthonormal basis there is only one way to express the function. This is what it means for a set of functions or vectors to be linearly independent.

Having a rational ratio of periods, $T_1/T_2=n/m$, means that the periods are $T_1=nT_0$ and $T_2=mT_0$ for some integers $n$, $m$. We can suppose that $n$ and $m$ have no common integer factor …

The matrix eigenvalue problem leads a recurrence equations with constant coefficients $$ x_{n+1}-2 x_n +x_{n-1}= \Omega^2 x_n, \quad x_{N+1}= x_1 $$ for the periodic case, and $$ x_{n+1}-2 x_n +x_{n …

I have not attempted to do the algebra, but I presume that you have found that the coeffeicient of $(\omega^2)^3$ in your characteristic equation is zero. To understand what this means consider the …